Optical systems are widely used in a variety of applications. One such application is the utilization of an optical system in a digital scanner or light lens copier. The optical system in a light lens copier is used to focus or direct light from a light source to a document or to focus or direct light reflected or transmitted from the document to a photoreceptor. On the other hand, an optical system in a digital scanner is used to focus or direct light from a light source to a document or to focus or direct light reflected or transmitted from the document to a CCD sensor device or full width array sensor. In carrying out this process, it is critical that the optical system directs the light to its destination without any errors.
A common error associated with optical systems is chromatic aberration. Chromatic aberration is classically defined in terms of paraxial ray intercept differences in light of two different wavelengths. This is illustrated in FIG. 1, which shows that axial color, involving the marginal ray, can be expressed either as a longitudinal error (.DELTA.f) or a transverse error (.DELTA.y). Lateral color, involving the chief ray, is defined as a transverse error (.DELTA.y). If the axial ray aberration is zero at two distinct wavelengths, the lens system is considered achromatic.
A mathematical way to look at chromatic aberration is to plot the axial and lateral color error as a function of wavelength as seen in FIG. 11. FIG. 11 shows transverse axial and lateral chromatic aberrations plotted as a function of wavelength .lambda. for a Kingslake Telephoto lens system using normal glasses. As can be seen from this Figure, the primary axial color correction is defined by the two widely spaced wavelengths. Moreover, it is noted that the maximum ray aberration error found at the intermediate wavelengths is defined as secondary spectrum.
The system shown in FIG. 11 has been designed to reduce chromatic aberration in the classical sense. In the classical definition, color correction refers to the lens system being color corrected at specific wavelengths. Thus, since FIG. 11 shows no axial color error at the approximate wavelengths of 460 nanometers and 585 nanometers and no lateral color errors at the approximate wavelengths of 425 nanometers and 575 nanometers, the Kingslake Telephoto system of FIG. 11, by the classical definition, is color corrected. However, to realize a more versatile color corrected lens system, the plots of the axial and lateral color error should be relatively flat and track the horizontal origin. If the system corresponding to FIG. 11 was to be color corrected in a more versatile manner, the two plots would be relatively flat and tracking the horizontal origin. Therefore, a goal of lens designing is to realize a lens system that is color corrected in a very versatile manner.
One problem in designing a color corrected optical system is the control of chromatic aberration through the selection of optical materials. In the past, designing color corrected optical systems has been a manual calculation relying on trial and error and experience. What is desired is a solution wherein lens system design is less rigorous and does not require a large amount of experience to implement.
Conventionally, the problem has been addressed using the classic "P vs. V glass triangle" method and the P* vs. P** technique. These methods assume thin lenses are in contact as an approximation. However, both of these methods cannot handle a lens system with significant air spaces.
In another conventional approach, the index of refraction of the optical materials, the optical power of a lens or lens system, is used in the calculations to color correct the system. However, as is well known, the index of refraction, N, is nonlinear when expressed as a function of .lambda.. Therefore, a third approach has been used which utilizes a rational function of .lambda., a chromatic coordinate .omega.. By expressing N as a function of .omega., only two or three terms are needed to accurately model the refractive index variation, far fewer than if .lambda. was the expansion variable. This third approach is commonly known as the Sigler method.
The use of the chromatic coordinate, .omega., or the Sigler method, enables a method of glass selection for air spaced achromats by using an equivalent set of equations for primary and secondary color as a function of .omega.. In this method, the optical material constants are defined as .eta..sub.i =v.sub.i /(N.sub.0 -1). This method also involves the use of a weighted summation of glass "vectors" in .eta..sub.1 vs. .eta..sub.2 space, similar to the classic "P vs. V glass triangle" method, but this method is more powerful since .eta..sub.1 and .eta..sub.2 represent true Taylor series coefficients (using .omega. as the the expansion variable). In this method, .eta..sub.1 =.eta..sub.2 =0 signifies the point of full primary and secondary color correction. However, this designing method also uses approximations, which give erroneous results if the air spaces in the lens system are large or the chromatic residual of a lens is large.
As shown above, some of the conventional methods for selecting optical materials rely on manual calculations, trial and error, and/or experience. Moreover, each of these methods rely on approximations being made in the chromatic aberration correction technique. To avoid the manual aspects of these methods, computer-aided searches have been utilized. An example of a computer-aided search to identify pairs of optical materials suitable for designing color corrected lens doublets is disclosed in U.S. Pat. No. 5,210,646 to Mercado et al. The entire contents of U.S. Pat. No. 5,210,646 are hereby incorporated by reference.
Mercado et al. discloses a computer-aided method for selecting optical materials to be used in designing a color corrected lens system. The method assumes that system consists of thin lenses in contact. As with other methods, Mercado et al. uses the chromatic coordinate, .omega., in the calculations to select the proper glass materials for the lens system. (It is noted that glass materials, as applicable to the present invention, can be simple glasses, plastics, or any other optical materials that are typically used in designing lens systems.) Thus, given that ##EQU1## the definition of the refractive index N(.omega.) is EQU N(.omega.)=N.sub.0 +v.sub.1 .omega.+v.sub.2 .omega..sup.2 +v.sub.3 .omega..sup.3 +. . . , (2)
and the optical material constants v.sub.i and .eta..sub.1 are ##EQU2##
Given the equations above, Sigler utilized the chromatic coordinate and developed a method of glass selection for air spaced achromats which took the classic equation for axial color .DELTA.y: ##EQU3## and empirically converted it to an equivalent equation for primary and secondary color as a function of .omega. and .eta..sub.i ##EQU4##
The approach used by Mercado et al. and Sigler for axial color correction assumes that the total contribution to secondary axial color is equivalent to that of a thin lens system. The effect of this assumption is to: (a) neglect a lens' thickness, and (b) neglect the induced component.
This approach, however, does not lead to a method of general validity for identifying compatible combinations of optical materials for designing color corrected lens systems since the method cannot be readily applied to a thick lens system or lens systems with separated components.
Thus, the conventional methods for correcting optical systems for chromatic aberration have remained either rigorous "cut-and-try" methods, methods with approximations, or methods having no general applicability.